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JOURNAL OF RESEARCH of the National Bureau of Standards— D. Radio Propagation 
Vol. 65D, No. 1, January-February 1961 

Useful Radiation From an Underground Antenna 

Harold A. Wheeler 

(February 26, 1960) 

An underground antenna delivers power to the surrounding conductive medium, and a 
fraction of the power goes out as radiation above the surface. This fraction is denoted the 
"radiation efficiency." It is expressed in simple terms for two types of underground anten- 
nas. The first and simplest is a vertical loop in a submerged spherical radome. The second 
is a submerged horizontal insulated wire with each end connected to a ground electrode. 
In each case, the efficiency is the product of three simple factors: The first depending on the 
index of refraction between air and ground; the second proportional to the size (radius of 
the radome or length of the wire) ; the third giving the attenuation with depth. An example 
for 1 megacycle per second gives an efficiency of 0.0014 for an underground wire of specified 
dimensions. The radiation efficiency is applicable to sender or receiver. 



1 2 



For protection from hazards above the ground, an 
underground antenna may be considered for some 
purposes, such as communication over a moderate 
distance. In a sender, such an antenna radiates a 
small part of its power out of the ground into free 
space. In a receiver, it intercepts a small part of 
the wave power that penetrates into the ground. 
Usually the most favorable path between two anten- 
nas is in the space above ground, even if both of 
the antennas are underground. 

It is the purpose of this note to give some simple 
formulas for the radiation efficiency and some other 
properties of underground antennas of two simple 
forms. These are based on previous publications 
of R. K. Moore [1] 3 , J. R. Wait [la], and the writer [2 
to 6], which were directed to underwater antennas 
presenting essentially the same problems. 

By formulating the efficiency of radiation out of 
the ground, separate from other losses in the antenna 
and in propagation, the formulas are simplified and 
the contributing factors are made apparent. These 
formulas are used in computing the performance of 
an underground antenna to be described by L. E. 
Rawls [7]. They are also applicable to the trans- 
mission formulas for computing system performance, 
as proposed by C. R. Burrows [8], H. T. Friis [9,10], 
and K. A. Norton [11]. This "separation of the 
variables" is usually helpful in understanding the 
respective factors in performance. 

The radiation pattern of the cases here to be 
considered is that of a small vertical loop just above 
ground. The reference for evaluating radiation 
efficiency is a perfect small loop just above a perfect 
ground plane, as shown in figure 1(a). The power 
supplied to the loop is all radiated; this useful 
radiated power is symbolized by P r . In a sender, 
the loop image doubles the far field. In a receiver, 



i Contribution from Wheeler Laboratories, Great Neck, N.Y., and Develop- 
ment Engineering Corp., Leesburg, Va. 

2 Paper presented at Conference on the Propagation of ELF Radio Waves, 
Boulder, Colo., Jan. 26, I960. 

3 Figures in brackets indicate the literature references at the end of this paper. 



Pr 



o 



(a) 



o 

IMAGE 



W^ 




a < 8 << X/2tt 



Figure l.^A perfect small loop as a basis for expressing the 
radiation efficiency of an underground antenna. 

a. Loop over perfect ground plane. 

b. Loop in submerged radome. 

the total interception area (3/2 radiancircle) is only 
half associated with the actual loop above ground, 
so it has an interception area of only % radiancircle, 
as referred to the power density in a plane wave 
along the ground. 

The simplest case of an underground antenna is 
a small loop in a small spherical radome located 
underground at a certain depth, as shown in figure 
1 (b) . The radome, of radius a, is filled with perfect 
insulating material (such as air), and its center is 
submerged to the depth d below the surface. 

The relative properties of the ground and the 
space above (air) are indicated by the radianlengths 
in the respective mediums, 8 and \/2tt. Assuming 
that the ground behaves mainly as a conductor 
rather than a dielectric, the former (5) is equal to 



89 



the well-known skin depth. All properties are to be 
expressed simply in terms of ratios of the several 
" length' ' dimensions. 

The "index of refraction" may be defined as 



?i=A/27r5»l 



(1) 



which is the ratio of phase velocities in the respective 
mediums. It is assumed to be very great, which is 
conducive to propagation of a wave along the ground 
with little "wave tilt." 

Assuming a perfect loop as the radiator in figure 
1(b), the power out of the loop (P d ) goes into the 
ground, and a small part of it (P r ) is radiated above 
ground. The ratio of these two quantities is the 
radiation efficiency: 



Pr 



£=(¥)" (I) (--¥) 



immersion 

(I/O 



size 

(loss) 



depth 
(atten.) 



(2) 



The three factors have the indicated significance. 
The middle factor is peculiar to this type, reflecting 
the excess dissipation in the near magnetic field if the 
size of the radome is much less than the radiansphere 
in the ground. 

The same value of efficiency is applicable in a 
receiver by reciprocity. In fact, the efficiency is 
more easily derived for a receiver, since plane waves 
may be assumed above and below the surface. It is 
then necessary also to compute the resistance appear- 
ing in the loop as a result of the surrounding ground, 
and this is easily done for the small spherical radome. 
In a receiver, the "noise temperature" of atmospheric 
radio noise above the surface may be multiplied by 
this efficiency to give the apparent temperature of 
such noise reaching the radome interior. This 
apparent temperature may be added to the actual 
temperature in the ground around the radome. 

Figure 2 shows another type of underground 
antenna, for which the radiation efficiency andsome 
other properties will be evaluated. The radiator is a 
long insulated wire buried in the ground in a hori- 
zontal position. Each end is connected to a ground 
electrode. In a sender, the current in the wire is 
provided by a generator connected in series. 

For simplification, it is assumed that the wire has 
a small radius (r<<<5) and a length (I) much greater 
than the skin depth but much less than the wave- 
length in air (37r<5<7<A/7r); also that the depth (d) is 



Pr 



-^mr 



o o 

INSULATED WIRE 

I >> 8 — 



d > 8 




greater than the skin depth (as indicated). Each 
ground electrode is assumed to be "perfect," ap- 
proximating a large plane perpendicular to the wire. 
On the same basis as the preceding loop antenna, 
the long wire has the radiation efficiency: 



Pr 

Pa 


=&)' (Id ( 




size 
(gain) 



exp- 



2d> 



(3) 



As before, the middle factor is peculiar to this type, 
but here there is a gain instead of a loss, in proportion 
to the length of the wire. 

The resistance and reactance (R and X) of the 
underground wire, as derived by R. K. Moore [1], 
are expressed as follows: 



R=-C0fX l- 



i R c ^=47.1^=296^ ohms (4) 

o A A A 



Q=l/p=X/R=^n^-0.577] 



4, 0.794 6 
-0.577 ==-ln 

7T r 



Figure 2. A horizontal wire as an underground antenna. 



(5) 



The wave resistance of free space is represented by 
R c =377 ohms. 

The resistance is that of the ground return circuit. 
It is equal to the resistance of a cross section 25X25 
of ground conductivity, on the basis of uniform cur- 
rent density, as indicated in the diagram. This 
resistance is involved in evaluating the efficiency. 
If the wire depth is less than the skin depth, the 
resistance changes very little. In fact, an older 
derivation by J. R. Carson [12] shows that the 
resistance has the same value for the special case of 
a wire located on the surface. 

The reactance is that of a coaxial line comprising 
the wire in a circular shield of radius about 0.8 8. 
The reactance is given for a wire of perfect con- 
ductivity and complete skin effect. For an actual 
wire small enough to neglect the skin effect, add }{ 
in the brackets in (5). If the depth is less than the 
skin depth, the reactance increases slightly; if on 
the surface, add }{ in the brackets in (5) for this 
change. 

Implicit in the radiation efficiency is the effective 
height of the loop formed by the wire and the 
ground return. The radiation above ground is the 
same as if there were a loop of the following effective 
height, with its center at the depth of the wire: 



A=5/V& 



-(6) 



It is remarkable that this is valid for any depth of the 
wire below the surface. 



90 



An example of the long wire underground is 
described by the following dimensions and computed 
properties: 

Frequency /=1 Mc/s, 

wavelength X=300 m, 

ground conductivity. _ o-=0.01 mlio/m, 

skin depth 5=5 m, 

depth d=\ m, 

length Z=100 m, 

radius r=3 mm, 

index of refraction n= 10, 

effective height i=1.8m, 

resistance R=99 ohms, 

reactance X= 900 ohms, 

power factor jp=0.11, 

radiation efficiency _P r /P d = 0.00 14=— 28.5 db. 

In this example, for each 1 kw of power into the 
ground, only 1.4 w will end up in useful radiation 
above the surface. 

The efficiency can be increased by locating a 
number of wires (N) in parallel in a grid, and sup- 
plying all with equal currents to form a sort of 
"current sheet." If the wires are spaced more than 
the skin depth (or more than twice the skin depth if 
near the surface) the efficiency is multiplied by the 
number (N) approximately. 

Two practical problems have been ignored in the 
above. One is the power loss in the end connections. 
The other is the current variation along the wire. 

The ground electrode for each end connection 
should have enough surface to reduce its resistance 
much below the irreducible value of the ground 
return (given above). 

Some current variation along the wire is caused 
by the voltage applied to the capacitance to ground 
through the insulation. The voltage developed by 
the resistance is inevitable. However, the voltage 
developed by the reactance can be reduced by 
inserting series capacitors along the length. In the 
present case, for example, a capacitor of 50-ohms 
reactance (about y 2 R) may be inserted at intervals 
of 50-ohms wire reactance to tune out this reactance 
at one frequency. This would require 18 series 
capacitors in the wire. 

The straight long wire does not have the highest 
efficiency possible for its length. A narrow loop 
of the same length, enclosed in a cylindrical radome, 
has greater efficiency by virtue of its wire return 
instead of a ground return. However, this loop 



has a much smaller power factor, so it must be 
critically tuned and has a relatively narrow band- 
width. 

These formulas for radiation efficiency and other 
properties are intended to indicate the behavior 
of certain types of underground (or underwater) 
antennas. They are expressed in simple terms, and 
they fit into the usual computations of terminal 
circuits and point-to-point transmission. 

The earlier work of Carson [12] and Moore [1] 
has contributed much to this subject, as mentioned 
in the text. The recent studies have been made as 
an aid to practical designs being proposed by Devel- 
opmental Engineering Corp. 

References 

[1] R. K. Moore, The theory of radio communication between 
submerged submarines, Cornell Univ., thesis (June 
1951). 

[la] J. R. Wait, The magnetic dipole antenna immersed in a 
conducting medium, Proc. IRE 40, 1244 (1952). 

[2] H. A. Wheeler, Radio wave propagation formulas, 
Bazeltine Rept. L301W (May 11, 1942). (Trans- 
mission formula as a power ratio, interception area, 
radiancircle.) 

[3] H. A. Wheeler, Universal skin-effect chart for conducting 
materials, Electronics 25, 152 (1952). (Formulas 
and chart, including land and water, all frequencies.) 

[4] H. A. Wheeler, Fundamental limitations of a small VLF 
antenna for submarines, IRE Trans. AP-6, 123 
(1958). (Coil in spherical radome submerged in 
water, power interception area, power factor.) 

[5] H. A. Wheeler, The spherical coil as an inductor, shield 
or antenna, Proc. IRE 46, 1595 (1958). (Coil in 
spherical radome submerged in water.) 

[G] H. A. Wheeler, The radiansphere around a small antenna, 
Proc. IRE 47, 1325 (1959). 

[7] L. E. Rawls, A practical underground transmitting an- 
tenna, topic presented bv Developmental Kngineering 
Corp. in meeting on Jan. 26, 1960, at the NBS, 
Boulder, Colo. 

[8] C. R. Burrows, Radio propagation over plane earth — field 
strength curves, Bell System Tech. J. 16, 45 (1937). 
(Transmission formula as a power ratio.) 

[9] H. T. Friis, A note on a simple transmission formula, 
Proc. IRE 19, 254 (1946). (Transmission formula 
as a power ratio in terms of interception area.) 
[10] S. A. Schelkunoff and H. T. Friis, Antennas— Theory 
and practice (John Wiley & Sons, Inc., New York, 
N.Y., 1952). (Transmission formula as a power 
ratio, problems, pp. 300-1.) 
[11] K. A. Norton, Transmission loss in radio propagation — ■ 
II, NBS Tech. Note 12 (June 1959). (Power ratios.) 
[12] J. R. Carson, Wave propagation in overhead wires with 
ground return, Bell System Tech. J. 5, 539 (1926). 

(Paper 65D1-106) 



565004— .60- 



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